Prestructuring sparse matrices with dense rows and columns via null space methods

TL;DR

This paper introduces null space methods to prestructure sparse matrices with dense rows or columns, aiming to improve the efficiency of direct solvers by modifying matrix structure before solving.

Contribution

It presents a novel application of null space techniques to eliminate dense rows and columns in sparse matrices, enhancing solver performance.

Findings

Null space methods effectively reduce dense structures in sparse matrices.

Prestructuring improves direct solver efficiency and stability.

One-sided null space application simplifies elimination of dense rows or columns.

Abstract

Several applied problems may produce large sparse matrices with a small number of dense rows and/or columns, which can adversely affect the performance of commonly used direct solvers. By posing the problem as a saddle point system, an unconventional application of a null space method can be employed to eliminate dense rows and columns. The choice of null space basis is critical in retaining the overall sparse structure of the matrix. A one-sided application of the null space method is also presented to eliminate either dense rows or columns. These methods can be considered techniques that modify the nonzero structure of the matrix before employing a direct solver, and may result in improved direct solver performance.